# Why y is a function of x ?

by mahmoud2011
Tags: function
P: 88
 Quote by Fredrik The proof goes like this: Let t0 be the real number such that $x(t_0)=6$. By assumption, y satisfies $y(t)=\sqrt{100-x^2(t)}$ for all t in some open interval that contains t0. Denote the following functions by f,g and h respectively. \begin{align} & s\mapsto s^2\\ & s\mapsto 100-s\\ & s\mapsto \sqrt{s} \end{align} Since$y=h\circ g\circ f\circ x,$ x is differentiable at t0, f is differentiable at x(t0), g is differentiable at f(x(t0)), h is differentiable at g(f(x(t0))).the chain rule tells us that y is differentiable at t0.
yes , I know this proof but some it becomes very hard to solve the equation , for general how we can know that the equation is define y implicitly as a differentiable function of x
, is it good to me to take advanced course in Calculus to know how .

Thanks
Emeritus